Quasirandom groups

Abstract: Babai and S\'os have asked whether there exists a constant c>0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy=z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.

• The paper shows there is no constant lower bound for the size of the largest product-free subset in every finite group, resolving a question by Babai and Sós.
• It defines group quasirandomness by linking the absence of low-dimensional representations to the impossibility of large product-free subsets in the group.
• The findings provide a new framework for analyzing group structure via representation theory, impacting fields like combinatorics and theoretical computer science.

Analysis of "Quasirandom Groups" by W. T. Gowers

The paper "Quasirandom Groups" authored by W. T. Gowers addresses the mathematical question posed by Babai and Sós regarding the existence of large product-free subsets in finite groups. Specifically, it questions whether there is a constant $c > 0$ such that every finite group $G$ contains a product-free subset whose size is at least $c|G|$. Gowers provides a negative answer to this question by demonstrating that no such constant exists. Furthermore, the author introduces conditions under which it is impossible for a group to have large product-free subsets, thereby identifying a connection with quasirandom properties of groups.

Key Contributions and Results

• Negative Resolution to Babai and Sós's Question: The paper conclusively negates the possibility that every finite group contains a large enough product-free subset as posed by Babai and Sós. Specifically, it is shown that for sufficiently large finite simple groups such as $\text{PSL}_2(q)$, the size of the largest product-free subset scales less than linearly, with bounds indicated mathematically as less than $Cn^{8/9}$.
• Defining Quasirandom Groups: Gowers defines a notion of quasirandomness for groups, correlating the lack of small-dimensional representations with quasirandom properties. This notion is based on equivalence between a group having no non-trivial low-dimensional representations and certain associative graphs being quasirandom.
• Connections to Representation Theory: Gowers leverages properties concerning irreducible representations to identify groups like $\text{PSL}_2(q)$ as possessing quasirandomness attributes. He uses the fact that these groups have no non-trivial irreducible representations of dimension lower than some bound to prove that they do not contain a large product-free subset.
• Exploring Product-Free Sets: By analyzing product-free subsets through the lens of representation theory, Gowers bridges group theory and graph theoretical notions of quasirandomness. This exploration further reveals insights into which groups can or cannot host dense product-free subsets.
• Extending Quasirandomness Characterizations: The paper not only focuses on the lack of large product-free subsets but extends the characterization by showing equivalencies with having large quotient groups being non-Abelian, and exponential/logarithmic dependencies of certain representation bounds.

Implications and Future Directions

The deduction that certain families of simple groups inherently possess quasirandom properties has implications for various fields, including combinatorics, number theory, and theoretical computer science. It contributes a rigorous framework for analyzing group structures in terms of their representations, potentially leading to further insights into the nature of these mathematical objects.

Theoretical Implications: The equivalence established between the absence of low-dimensional representations and graphs associated with a group's elements being quasirandom opens avenues for deeper exploration of group symmetry and randomness. There is potential for developing further group-theoretical criteria akin to quasirandomness to classify objects in other mathematical domains.

Practical Implications: The results could influence algorithms in computational group theory, especially in tasks concerning subgroup identification and representation finding. The concept of using representation dimension as a constraint could streamline approaches in automation and complexity evaluations in computational settings.

Future Research Directions: Gowers hints at questions concerning stronger bounds for product-free subset sizes and the detailed spectral properties of related graphs. Investigating other families of groups for similar quasirandom properties or extending these notions to infinite or non-finite settings could form an intriguing continuation of this work.

In summary, this extensive treatise on quasirandom groups by Gowers not only resolves significant combinatorial questions but also sets a sophisticated stage for ongoing inquiry into the quasirandom properties of mathematical structures. The depth of analysis and connection across disciplines reflects the paper’s potential to provoke further theoretical developments and applications.